Probability math

Question

Every mushroom is edible with the probabiloity 0.95, independent of other picked mushrooms.

What would be the the probability that 300 mushrooms picked under a month are edible amongst 320 picked?

How many mushrooms would have had to be picked so that the probability that at least 300 would be edible would be at least 0.99?

Answer 1

P(mushroom is edible) = 95%

pick 320, what is probability 300 are edible?

300/320= 30/32= 15/16= 93.75%

you'd expect about .95(320) = 304 edible mushrooms if picked 320

50% chance 304 or more

50% chance 304 or less

to get 300 or more is greater than 50%

use a binomial distribution, as there are just 2 possibilities, edible or not edible (no inbetween or percentages of edible) binominal approximates a normal distribution

mean =np- 320x.95= 304

var=npq = 320(.95)(.05)=15,2,

s=sqr(npq)= sqr15,2 = about 3.898717738= about 3.9

calculate the z score and find corresponding area from z tables

z=(x-xbar)/s =about (300-304)/3.9= about -1.026

= about -1

by the empirical rule 68% are within 1 s of the mean

(1-.68)/2= .32/2 = .16=16% below 300, 84% above 300

for a more precise calculation use z tables or a z calculator

which is:

15.245% less than 300

84.755% more than 300

how many do you need to pick to get 300 edible with 99% probability?

another question

more calculations

by the empirical rule, an approximation that's 2 s or more precisely 1.98 standard deviations from the mean

then

calculate the z scores again, and corresponding area

Answer 2

The probability that exactly 300 of the 320 mushrooms are edible is 0.05625; the probability that at least 300 of the 320 mushrooms are edible is 0.87392.

326 is the smallest number of trials that will yield at least 300 edible mushrooms with a 99% or better probability.

I used 2 different online binomial probability calculators and reached the same results with both systems.

P(X) = _nC_x * p^xq^n-x

In this case, p = 0.95, q = 1 - p, x = 320, n = 20, etc.